Identification in Bayesian Estimation of the Skewness Matrix in a Multivariate Skew-Elliptical Distribution

TL;DR

This paper improves Bayesian estimation of the skewness matrix in multivariate skew-elliptical distributions by addressing label switching issues through a lower-triangular constraint and introducing a sparse estimation approach with the horseshoe prior.

Contribution

It proposes a modified sampling algorithm with an identification constraint and a Bayesian sparse estimation method to accurately estimate the skewness matrix.

Findings

Successful estimation of the true skewness structure in simulations

Overcomes label switching issues in Bayesian estimation

Enhanced interpretability of the skewness matrix

Abstract

Harvey et al. (2010) extended the Bayesian estimation method by Sahu et al. (2003) to a multivariate skew-elliptical distribution with a general skewness matrix, and applied it to Bayesian portfolio optimization with higher moments. Although their method is epochal in the sense that it can handle the skewness dependency among asset returns and incorporate higher moments into portfolio optimization, it cannot identify all elements in the skewness matrix due to label switching in the Gibbs sampler. To deal with this identification issue, we propose to modify their sampling algorithm by imposing a positive lower-triangular constraint on the skewness matrix of the multivariate skew- elliptical distribution and improved interpretability. Furthermore, we propose a Bayesian sparse estimation of the skewness matrix with the horseshoe prior to further improve the accuracy. In the simulation study, we demonstrate that the proposed method with the identification constraint can successfully estimate the true structure of the skewness dependency while the existing method suffers from the identification issue.